Sequences and the nth Term: O-Level / IGCSE Maths (0580)

A sequence is an ordered list of numbers called terms, each generated by a rule. This topic looks at how to spot that rule, continue a sequence by a few more terms, and, most usefully, find a formula for the $n$th term so that any term can be calculated directly without listing all the ones before it. The syllabus focuses on two families: linear sequences, where the terms go up (or down) by a constant amount, and simple quadratic sequences, where the gap between terms itself changes by a constant amount.

For a linear sequence, the constant gap between consecutive terms is the common difference $d$. The $n$th term has the form $u_n = dn + c$: multiply the position $n$ by $d$, then add the constant $c$ needed to match the first term. For example, $5, 8, 11, 14, \dots$ has $d = 3$, giving $u_n = 3n + 2$, so the 50th term is $3(50)+2 = 152$. For a quadratic sequence, the first differences are not constant but the second differences are; halving that constant second difference gives the coefficient $a$ in $u_n = an^2 + bn + c$, and the remaining constants are found by comparing the $an^2$ values with the original terms. Checking that your formula reproduces the given terms is a quick and reliable way to confirm an answer.

The worked examples below are original, written from the syllabus objective to show these exam-style methods step by step. Each worked solution sets out how to identify the type of sequence, build the $n$th-term rule, and use it to continue the pattern or evaluate a specific term.